June  2017, 7(2): 201-210. doi: 10.3934/naco.2017014

Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of the Russian Academy of Sciences, 134, Lermontov St., 664033, Irkutsk, Russia

* Corresponding author: Stepan Sorokin

Received  December 2016 Revised  May 2017 Published  June 2017

Fund Project: This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

We consider a class of rightpoint-constrained state-linear (but non convex) optimal control problems, which takes its origin in the impulsive control framework. The main issue is a strengthening of the Pontryagin Maximum Principle for the addressed problem. Towards this goal, we adapt the approach, based on feedback control variations due to V.A. Dykhta [4,5,6,7]. Our necessary optimality condition, named the feedback maximum principle, is expressed completely in terms of the classical Maximum Principle, but is shown to discard non-optimal extrema. As a connected result, we derive a certain form of duality for the considered problem, and propose the dual version of the proved necessary optimality condition.

Citation: Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
References:
[1]

A. ArutyunovD. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688. 

[2]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.

[3]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.

[4]

V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396. 

[5]

V. A. Dykhta, Positional strengthenings of the maximum principle and sufficient optimality conditions, Proceedings of the Steklov Institute of Mathematics, 293 (2016), S43-S57. 

[6]

V. A. Dykhta, Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls, Autom. Remote Control, 75 (2014), 829-844.  doi: 10.1134/S0005117914050038.

[7]

V. A. Dykhta, Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems, Autom. Remote Control, 75 (2014), 1906-1921.  doi: 10.1134/S0005117914110022.

[8]

V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.

[9]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides: Control System, Kluwer Acad. Publ., 1988. doi: 10.1007/978-94-015-7793-9.

[10]

V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.

[11]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.

[12]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.

[13]

V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980.

[14]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4615-0095-7.

[15]

M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. 

[16]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.

[17]

R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures}, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. 

[18]

O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418.  doi: 10.1134/S0005117915030054.

[19]

S. P. Sorokin, Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization, Autom. Remote Control, 75 (2014), 1556-1564.  doi: 10.1134/S0005117914090021.

[20]

A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003.

[21]

J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438. 

[22]

S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.

show all references

References:
[1]

A. ArutyunovD. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688. 

[2]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.

[3]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.

[4]

V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396. 

[5]

V. A. Dykhta, Positional strengthenings of the maximum principle and sufficient optimality conditions, Proceedings of the Steklov Institute of Mathematics, 293 (2016), S43-S57. 

[6]

V. A. Dykhta, Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls, Autom. Remote Control, 75 (2014), 829-844.  doi: 10.1134/S0005117914050038.

[7]

V. A. Dykhta, Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems, Autom. Remote Control, 75 (2014), 1906-1921.  doi: 10.1134/S0005117914110022.

[8]

V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.

[9]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides: Control System, Kluwer Acad. Publ., 1988. doi: 10.1007/978-94-015-7793-9.

[10]

V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.

[11]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.

[12]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.

[13]

V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980.

[14]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4615-0095-7.

[15]

M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. 

[16]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.

[17]

R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures}, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. 

[18]

O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418.  doi: 10.1134/S0005117915030054.

[19]

S. P. Sorokin, Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization, Autom. Remote Control, 75 (2014), 1556-1564.  doi: 10.1134/S0005117914090021.

[20]

A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003.

[21]

J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438. 

[22]

S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.

[1]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[2]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[3]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174

[4]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[5]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110

[6]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031

[7]

C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial and Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435

[8]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[9]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022

[10]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275

[11]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[12]

Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067

[13]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential. Evolution Equations and Control Theory, 2017, 6 (1) : 35-58. doi: 10.3934/eect.2017003

[14]

Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial and Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591

[15]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[16]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[17]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041

[18]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311

[19]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[20]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations and Control Theory, 2022, 11 (1) : 239-258. doi: 10.3934/eect.2021001

 Impact Factor: 

Metrics

  • PDF downloads (180)
  • HTML views (212)
  • Cited by (0)

Other articles
by authors

[Back to Top]